Gts original k independent pair master quadratic model

In the local-spectral branch write $X=\mathrm{Tot}(K_Y)$ with $Y$ a smooth separated quasi-projective threefold, $A={\mathrm{Sym}}_{{\mathcal{O}}_Y}{K}_Y^{-1}$ and projection $\pi$. A compactly supported coherent $F$ is an $A$-module $G={\pi }_{\ast }F$ with action $m:K_Y^{-1}\otimes G\to G$, equivalently $\phi :G\to G\otimes K_Y$. The support map to $Y$ is affine and proper, hence finite, so $G$ is coherent. These assertions hold relatively for flat families with proper relative support.

For the specified finite CY4 dg-quiver, keep the original potential $H$ and add separate primary and secondary vertices. The secondary source maps to every original vertex and to the primary vertex, with distinct gauges. Give every added degree-zero arrow its opposite degree-minus-two arrow and each new vertex its degree-minus-three loop. For the covector relation add the degree-minus-one pair $\rho ,{\rho }^{\ast }$ and the term ${\rho }^{\ast }bc$ in the original cyclic order. The old potential uses none of these variables. The new term uses neither $\rho$ nor the new degree-minus-two opposites, so each paired contribution to its self Poisson bracket and each cross bracket with $H$ is zero. Thus the original master equation remains true. Its cyclic derivatives impose exactly $bc=0$ and the indicated dual differentials. This proves the relation-pair input rather than postulating a new potential convention.

There is a functorial induced-module resolution $0⟶A\otimes (G\otimes K_Y^{-1})\stackrel{a\otimes (g\otimes l)\mapsto al\otimes g-a\otimes m(l\otimes g)}{\to }A\otimes G⟶G⟶0.$ Locally trivialize $K_Y^{-1}$. The first map is $t-m$; its highest polynomial coefficient proves injectivity without a flatness assumption on $G$. Repeated division by $t-m$ reduces any polynomial to its prescribed action on $G$, proving exactness in the middle and at the last term. Changing the line trivialization respects both terms, so the resolution glues.

Resolve the $G_i$ locally by finite locally free complexes and use the induced resolution. Derived adjunction gives the natural fibre sequence $R{\mathrm{Hom}}_X(F_1,F_2)⟶R{\mathrm{Hom}}_Y(G_1,G_2)\stackrel{{\phi }_2(-)-((-)\otimes 1){\phi }_1}{\to }R{\mathrm{Hom}}_Y(G_1,G_2\otimes K_Y)\mathrm{.}$ For this computation choose a finite affine open cover of the proper support inside the separated scheme $Y$. Every finite intersection is affine, and the sheaf-Hom terms from the locally free resolutions are quasi-coherent, hence acyclic on those intersections. The fixed-cover Čech-to-sheaf comparison is therefore a quasi-isomorphism and its total complex computes derived global sections. For a family, use the same cover after restricting the base to an affine open; on each affine intersection, derived global sections commute with derived base change. Refinement maps and changes of locally free resolutions induce the canonical comparison maps, so the relative fibre sequence is independent of these choices. Thus this is an actual derived fibre sequence, including the specified zero-composite homotopy, not merely an Euler-characteristic identity.

The needed compact-support duality on $Y$ can be read from the GIVEN geometric CY4 trace on $X$, so it is not an additional unproved spectral input. Apply step 3.1 to the zero-Higgs modules $i_{\ast }{G}_j$ on the zero section. The commutator is zero, so its two summands are $R{\mathrm{Hom}}_Y(G_1,G_2)$ and $R{\mathrm{Hom}}_Y(G_1,G_2\otimes K_Y)[-1]$. Fibre dilation separates them with weights zero and one. In a trivialization the CY4 volume is $d{y}_{1}d{y}_{2}d{y}_{3}dz$ and has fibre weight one; hence the CY4 trace pairs these two summands, not either summand with itself. Its perfectness identifies the second summand with $R{\mathrm{Hom}}_Y(G_2,G_1{)}^{\vee }[-4]$. Thus the last term of step 3.1 is $R{\mathrm{Hom}}_Y(G_2,G_1{)}^{\vee }[-3]$. The trace identity for the two commutators follows by moving $\phi$ cyclically past the homogeneous cochains with their Koszul signs; the terms cancel in pairs. Consequently the fibre sequence is self-dual. For $F_1=F_2$, dualizing and shifting by $[-1]$ gives the global triangle $C⟶E⟶C^{\vee }[2],C=R{\mathrm{Hom}}_Y(G,G{)}^{\vee }[-1],E=R{\mathrm{Hom}}_X(F,F{)}^{\vee }[-1]\mathrm{.}$ Its homotopy is the fibre-sequence homotopy of step 3.1. Equivariant resolutions and the trace produce the same triangle equivariantly.

We verify the geometric category underlying this construction instead of assigning it by its classical truncation. Let $\mathcal{D}$ be a dg model of $\mathrm{Perf}(Y)$, with bimodules represented by diagonal kernels. The diagonal of the smooth threefold has the coordinate Koszul resolution $d{e}_i=y_i-x_i$. Dualizing its top exterior term contributes $\det T_Y=K_Y^{-1}$, so its inverse dualizing kernel is ${\mathrm{\Delta }}_{\ast }{K}_Y^{-1}[-3]$. In its four-dimensional tensor completion the shifted generator is ${\mathrm{\Delta }}_{\ast }{K}_Y^{-1}$ in degree zero. Its $n$-fold convolution is ${\mathrm{\Delta }}_{\ast }{K}_Y^{-n}$, with the ordinary tensor multiplication. The resulting algebra is exactly ${⨁}_{n\ge 0}{K}_Y^{-n}=\mathrm{Sym}{K}_Y^{-1}$, since this is one invertible generator. Relative Spec and the actual induced $t-m$ resolution therefore identify its old modules and their Hom differential with the actual spectral sheaves on $X$, including the CY determinant. This is an explicit old-sector comparison, not an assumption that an arbitrary ${\mathcal{B}}_k$ has that sector.

The original primary pair is $Q=(L_p,E,s_p:L_p\to {\mathrm{F}\mathrm{r}}_k(E))$, with primary source degree zero or one. On $Y$ use the same actual primary decoration of $G$. On the original concentration open its framing complex is the stated vector bundle. If its cotangent boundary is $A_p\to C$, its self-dual reduction is $E_p=\mathrm{cofib}\text{\hspace{0.17em}⁣}(A_p⟶\mathrm{fib}(E⟶A_p^{\vee }[2]))\mathrm{.}$ The cotangent triangle sets $C_p=\mathrm{cofib}(A_p\to C)$. Taking the fibre/cofibre $3\times 3$ diagram gives $C_p\to E_p\to C_p^{\vee }[2]$, with the inherited global homotopy. Retain the fixed rank-one summand and apply the source trace-free projection throughout; it commutes with all maps fixing that summand. This is the actual pair complex, not a reidentification of its source with an auxiliary line.

Distinguish the actual classical diagram space from its CY4 enhancement: the latter includes the negative-degree dual coordinates and is not the naive derived stack of arrows alone. The classical $X$-diagram is the spectral pullback of the classical $Y$-diagram. This follows on every derived base from a free-algebra argument, not only on tangent spaces. The algebra $A=\mathrm{Sym}{K}_Y^{-1}$ is also the tensor algebra on this one invertible degree-zero generator. An $A$-action on a perfect $G$ is therefore precisely a map $K_Y^{-1}\otimes G\to G$, including its derived homotopies; all its iterates are forced, with no additional relation. A map from a free framing module $A\otimes L_k\otimes V$ to that module is, by derived adjunction, exactly a map $L_k\otimes V\to G$, with no equation $\phi s=0$. This computes the derived spectral and framing deformation functors on arbitrary Artin thickenings. Relative Serre duality identifies the Higgs-map complex with the shifted cotangent fibre used in step 4.1.

We construct the geometric source cross-duality actually used here. Put $B=R{\mathrm{Hom}}_X(F,F)$, $M=R{\mathrm{Hom}}_X(\mathcal{O},F)$ and $N=R{\mathrm{Hom}}_X(F,\mathcal{O})$ for the actual source line $\mathcal{O}={\pi }^{\ast }{L}_Y$. The given trace for compact objects alone does not apply to the noncompact line. Instead embed the smooth quasi-projective $X$ in an open of projective space and push its proper-support sheaf to that projective ambient. On its standard cover, the Laurent-monomial Čech contraction pairs the coefficient of $x_0^{-1}\cdots x_m^{-1}$ with complementary monomials, over any coefficient algebra. The diagonal Koszul resolution extends this trace from twists to the supported complex: its successive differences $x_i-y_i$ are monic in a new first-factor coordinate, hence regular after tensoring any second-factor module. Dualizing the regular-immersion Koszul resolution contributes the top determinant $K_X[4]$; the actual CY trivialization identifies it with ${\mathcal{O}}_X[4]$. The coordinate determinant changes cancel, so this constructs the evaluation pairing $N\simeq M^{\vee }[-4]$ globally. The finite relative resolutions used here exist over singular bases: repeatedly surject sums of projective twists onto the flat supported sheaf and its flat kernels. After the fibre's projective homological-dimension bound, the last kernel has locally free fibres; lifting a fibre basis, base flatness kills ${\mathrm{Tor}}_1$ and Nakayama kills the presentation kernel, so it is locally free near that fibre. Common refinements compute the same trace and commute with derived base change. This proves the needed source cross-pairing, with all equivariant CY weights retained.

Here is the actual primary geometric cochain map. Use the complex $I=[\mathcal{O}\stackrel{s}{\to }F]$ with $\mathcal{O}$ in degree zero and a perfect resolution $P[-1]$ of its degree-one term. Its homogeneous endomorphism differential is $h\mapsto Dh-(-1{)}^{\mathrm{\mid }h\mathrm{\mid }}hD$, for the actual cone differential $D$. Write the four blocks as $(a,b,u,v)$ in $R\mathrm{Hom}(\mathcal{O},\mathcal{O})$, $B$, $M[-1]$ and $N[1]$. Supertrace is $a-{\mathrm{tr}}_P(b)$, so the trace-free homotopy fibre is represented by the three-block complex via $q(a,b,u,v)=(b,u,v)$ and $j(b,u,v)=({\mathrm{tr}}_P(b),b,u,v)$. Its differential is the displayed matrix commutator followed by that substitution: the reverse block contributes the diagonal composites $vs$ and $sv$, and the old diagonal contributes $s{\mathrm{tr}}_P(b)-(-1{)}^{\mathrm{\mid }b\mathrm{\mid }}bs$ in the forward block. The identity $\mathrm{str}[D,h]=0$ shows this is a subcomplex and $j$ is a chain map; $qj$ and $jq$ are identities. On changes of resolutions use the homotopy fibre and the actual trace comparisons rather than unrelated strict kernels. Dualizing $q$ and shifting by $[-1]$, then using step 5.3, identifies the actual geometric obstruction complex $R\mathrm{Hom}(I,I{)}_0^{\vee }[-1]$ with $E\oplus M^{\vee }\oplus M[2]$. Its differential is exactly the primary fibre/cofibre differential: the first boundary is the shifted dual of $b\mapsto bs-s{\mathrm{tr}}_P(b)$, its other boundary is its trace-cyclic adjoint, and the corner homotopy is the cone homotopy from $D^2=0$. Thus $q^{\vee }[-1]$ and $j^{\vee }[-1]$ are the required inverse primary comparisons to $E_p$, preserving evaluation and the specified trace-free factor.

Define the secondary bundle on the primary pair stack by $V_{+}(Q)={\mathrm{F}\mathrm{r}}_k(E)\oplus L_p\mathrm{.}$ A new source $L_a$ has independent maps $f_a:L_a\to {\mathrm{F}\mathrm{r}}_k(E)$ and $g_a:L_a\to L_p$. The primary $s_p$ remains separate and there is no equation $f_a=s_p{g}_a$. The two source gauges are distinct. This is an actual finite vector-bundle decoration, exact on the stipulated triples, with dimension $f_k(E)+\dim L_p$; it needs no new divisor or cofinal-index hypothesis.

The geometric primary obstruction arrows agree as maps, not merely on ranks. On a square-zero extension lift the actual resolution differential and section and compute $(D+\delta {)}^2$. Its old diagonal is the sheaf curvature, its forward block is the evaluation-lift defect, and its reverse block is their trace-dual defect. The matrix substitutions in step 6.1 send these very same defects to the primary reduction differential. Changing a lift changes the defect by the same commutator coboundary. Universal derivation consequently gives a homotopy-commutative square between the actual cone obstruction arrow and $E_p\to L$ for every coefficient module. Trace, evaluation and the induced-module resolution commute with common Čech refinements; these actual maps and homotopies descend and commute with derived base change.

Replace $L_a$ by its positive flag chain. On the required injective locus the relative flag map over the primary $Y$ stack is smooth. For the covector master adjoin the sink $V_0$, $a:V_1\to V_0$ and $b:V_l\to V_0$, and impose $bc=0$ for $c:V_{l-1}\to V_l$. In a basis adapted to the injective $c$, these equations eliminate distinct coordinates of $b$. Their Koszul algebra resolves their zero locus over an arbitrary old base, including a derived base. The remaining $b$ and $a$ are vector-bundle coordinates before taking the auxiliary gauge quotient. On the full injective locus the actual relative cotangent is the perfect quotient complex $N_{+}=[{\mathrm{\Omega }}_{W\mathrm{/}P}\to {\mathfrak{g}}_{\mathrm{a}\mathrm{u}\mathrm{x}}^{\ast }\otimes {\mathcal{O}}_W]$ in degrees $0,1$, descended from its equivariant atlas $W$ over the old pair base $P$; the arrow is the dual infinitesimal action. It is a vector bundle in degree zero on the actual free locus, since there the infinitesimal action is injective and this dual map is surjective. Injectivity of $c$ alone does not remove a sink stabilizer: at $a=b=0$ the sink ${\mathbb{G}}_m$ remains, so its degree-one cotangent term must be retained. The construction keeps this full smooth Artin locus and only uses a degree-zero bundle when the actual free/stable locus has been stipulated.

The sink freeness used on a stipulated finite-phase semistable locus is a separate assertion. If the combined incoming map $(a,b)$ fails to span its sink, its cokernel is a nonzero pure-sink quotient. The kernel retains the entire old object and every positive source, and is therefore an admissible subobject with identity old-part map. Its phase is finite or positive-infinite, whereas the prescribed negative sink has negative-infinite phase. The weak stability inequality forbids this exact sequence. Thus $(a,b)$ spans the sink there. Injective positive flag composites force their auxiliary automorphisms fixing the old primary object to be the identity, and this spanning condition forces the sink automorphism to be the identity. The same argument kills infinitesimal endomorphisms. This proves freeness, and the degree-zero bundle conclusion, exactly on that stipulated locus. On the full injective Artin master keep the quotient cotangent complex of step 7.2.

For the stipulated auxiliary-weight stability, a pure auxiliary dimension vector $(n_0,n_1,\dots ,n_r)$ has numerator ${\mu }_0{n}_0+{\sum }_{i=1}^r{\mu }_i{n}_i$, with ${\mu }_i>0$ for positive vertices and ${\mu }_0<0$ at the sink; positive numerator has phase $+\mathrm{\infty }$, and a nonzero pure sink has phase $-\mathrm{\infty }$. Let $d_0=\dim V_0$ be its actual fixed dimension. The kernels $K_i$ of the positive flag composites into $V_{+}$ form a subrepresentation of the positive chain. Give its sink the span $K_0\subseteq V_0$ of the incoming images from these kernels. It is then a pure-auxiliary subobject, satisfying the restricted relation $bc=0$. If any $K_i\mathrm{\ne }0$, its numerator is at least ${\min }_i{\mu }_i-d_0\mathrm{\mid }{\mu }_0\mathrm{\mid }$. Choose the negative weight in the very-small chamber with $0<\mathrm{\mid }{\mu }_0\mathrm{\mid }<{\min }_i{\mu }_i\mathrm{/}\max (1,d_0)$. The numerator is strictly positive, so this subobject has phase $+\mathrm{\infty }$ and destabilizes a finite-phase object: by weak see-saw its nonzero quotient must have finite or smaller phase. Thus every positive composite is injective. The choice accommodates every stated sink dimension, including $d_0=0$, and uses no ordinary-kernel closure in the old weak category.

The secondary cotangent triangle gives $u_Y:A_{+}=N_{+}[-1]\to C_p$. Compose with step 5.1. Its shifted dual factors through $E_p\to C_p^{\vee }[2]$, so $A_{+}⟶E_p⟶A_{+}^{\vee }[2]$ has the global self-dual zero-composite homotopy induced by that triangle. Define $K_{+}=\mathrm{fib}(E_p\to A_{+}^{\vee }[2]),E_{+}=\mathrm{cofib}(A_{+}\to K_{+})\mathrm{.}$ Dualizing this fibre/cofibre diagram proves the prescribed shifted self-duality, including the actual differentials. Its virtual rank is $\mathrm{rk}{E}_p+2\mathrm{rk}{N}_{+}$, with the quotient-complex rank on the Artin locus and the bundle rank on the free locus.

The sheaf nullhomotopy and the secondary obstruction arrow come from the actual spectral cone, not a locally chosen isotropic extension. Model the old Hom by the induced $t-\phi$ resolution and the primary by the actual cone of step 6.1. The augmentation to the $Y$ column kills its other cone coordinate; its composite with the cone inclusion is strictly zero with its canonical cone homotopy. Compose that homotopy with the actual cotangent boundary $u_Y$ and its trace-cyclic dual. This gives the specified homotopy of $A_{+}\to E_p\to A_{+}^{\vee }[2]$. On a square-zero lift of the independent arrow and relation generators, the new errors are the actual arrow-equivariance defects and $\delta bc+b\delta c$ in the relation row. Those are the rows of $u_Y$, its shifted adjoint and this cone homotopy. Eliminating $(b{\mathrm{\mid }}_{\mathrm{im}c},r)$ contracts their error rows and their dual rows together. Universal derivation of this total complex therefore is the actual secondary reduction arrow $E_{+}\to L_{Z_X}$, compatible with the primary arrow and every coefficient module. This proves the obstruction-arrow comparison for the constructed geometric reduction; it asserts no identification with an arbitrary separately specified enhancement.

Here is the explicit CY4 enhancement of that actual diagram. On a semi-free derived chart $(A_Y,d)$ of the $Y$-diagram stack write $d{x}_i=f_i(x)$. Adjoin a shifted dual coordinate ${\xi }_i$ of degree $-2-\mathrm{\mid }{x}_i\mathrm{\mid }$. The cotangent Poisson bracket has degree two. Define the Hamiltonian $H_d$ as the tautological pairing of $\xi$ with $d$, with Koszul signs fixed by $\{H_d,x_i\}=f_i$. Its other equations are the shifted transposes of the Jacobian $\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{x}_i$. The identity $\{H_d,H_d\}=0$ is exactly $d^2{x}_i=0$ for all $i$, so this is an actual differential graded algebra, not just a quadratic form on a tangent complex. Apply the construction AFTER adjoining the independent arrow coordinates and the Koszul generator $r$ with $dr=bc$. Then $\mathrm{\partial }(bc)=\delta bc+b\delta c$ is the actual relation differential, and there is still no relation $f_a=s_p{g}_a$. Source gauge terms and their shifted duals are included before taking the rigidified quotient. These chart algebras define the secondary CY4 derived enhancement. They glue by the cotangent coordinate rule: under change of semi-free resolution the pairing with the derivation and the Liouville form are natural. Thus they give the global shifted-cotangent completion of the actual derived $Y$ decoration.

Adjoin the named formal sources using the directed right-module convention $\mathrm{Hom}(P,p)=R{\mathrm{Hom}}_Y(P,L_Y)$, $\mathrm{Hom}(p,P)=0$, retaining primary and auxiliary labels. Its diagonal has the old diagonal and the universal derivative rows of the directed generators, with boundary $z\otimes 1-1\otimes z$ at each endpoint. Use the FULL normalized relative bar before taking a finite perfect representative and its derived dual. The word-cut contraction in11.2 proves exactness and keeps every row $d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}(dz)$, including all higher syzygies and $d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}r=(d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}b)c+b(d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}c)$. Dualizing this actual row, including its boundary, gives its opposite and moment terms. Evaluation of a homogeneous coevaluation is ${\sum }_{\alpha }(-1{)}^{\mathrm{\mid }{e}_{\alpha }\mathrm{\mid }}{e}_{\alpha }\otimes e_{\alpha }^{\vee }$; hence both endpoint compositions and the actual supertrace occur, with their Koszul signs. To bind this calculation to the ACTUAL completion, put $\mathcal{B}=T_{{\mathcal{D}}_{+}}(\theta )$, $\theta ={\mathrm{\Theta }}_{{\mathcal{D}}_{+}}[3]$. For its actual module $M$ on the original family use the relative induced resolution $M{\otimes }_{{\mathcal{D}}_{+}}\theta {\otimes }_{{\mathcal{D}}_{+}}\mathcal{B}\stackrel{\alpha }{\to }M{\otimes }_{{\mathcal{D}}_{+}}\mathcal{B},\alpha (m\otimes x\otimes b)=mx\otimes b-m\otimes xb\mathrm{.}$ After a cofibrant module replacement its cone resolves $M$: expanding in tensor words in $\theta$, the last-word cut contracts the augmented kernel, including a NONZERO old action. Applying Hom gives the actual completed endomorphism complex as the fibre of $R{\mathrm{Hom}}_{{\mathcal{D}}_{+}}(M,M)\to R{\mathrm{Hom}}_{{\mathcal{D}}_{+}}(M\otimes \theta ,M)$, whose map is that left/right action commutator. Resolve the diagonal before taking its dual; finite tensor-Hom evaluation identifies the second term with $R{\mathrm{Hom}}_{{\mathcal{D}}_{+}}(M,M{)}^{\vee }[-3]$. The identification is the Casimir/coeval sum just displayed, and is valid here because the old support and all source spaces are proper/finite. Thus the opposite rows and endpoint map in this Hom fibre are precisely the duals of the FULL directed bar rows. This proves that the uncontracted complex used in11.2, 12.2, 13.2 and 14.1 computes the specified inverse-diagonal completion, rather than a new complex assigned by its ranks. Its comparison is the word-cut contraction and finite section perturbation below, not identity on a supposed old completed dg subcategory.

Keep the original base $Z_X$ and its actual universal geometric diagram; its full cotangent complex is not replaced or truncated here. Its prescribed ordinary forgetful maps forget the named forward data. To evaluate the constructed completed diagonal on this family, use the dg QUOTIENT killing the newly adjoined inverse-diagonal dual rows and source moment loops, while retaining the old spectral algebra and the forward extension. This ideal is stable: the old differentials do not involve the new forward generators, and differentiating a new forward equation introduces only a new dual row in the dual differential; every added endpoint contribution to an old moment contains such a new dual. The quotient therefore has the actual old algebra and its free forward framing bimodules, with the resolved $bc$ relation. Restriction of scalars along this quotient assigns a completed module to the actual universal diagram, and commutes with every derived pullback of that diagram to $Z_X$. This constructs the evaluated cotangent comparison on the ORIGINAL base. It is not a functor in the reverse direction on all independently chosen derived completed-module points. Opposite and moment rows are retained in the FULL evaluated diagonal even when their universal-module values are zero; the comparison never removes their infinitesimal or obstruction contributions. The actual cochain comparison and cotangent square are proved in11.2, 12.2, 13.2, 14.1, 15.1, 16.1 and 17.1. Thus neither the base nor the class of completed modules is silently changed to a smaller zero-dual moduli functor.

We give the bar contraction explicitly. Locally trivialize $K_Y$ and write $A=R[t]$, $A^e=R[t_L,t_R]$. The relative diagonal is $K=[A^{e}e\to A^e]$ with $de=t_L-t_R$. On the normalized bar set $p_0=1$, $p_n=0$ for $n\ge 2$ and

For $n\ge 1$ put $h_n[t^{a_1}\mathrm{\mid }\cdots \mathrm{\mid }{t}^{a_n}]=(-1{)}^n{\sum }_{j=1}^{a_n-1}[t^{a_1}\mathrm{\mid }\cdots \mathrm{\mid }{t}^{a_{n-1}}\mathrm{\mid }{t}^j\mathrm{\mid }t]t_R^{a_n-1-j}\mathrm{.}$ The last multiplication faces telescope; all earlier faces cancel with $hd$, leaving $dh+hd=1-ip$. Every output ends in the single letter $t$, proving $h^2=hi=ph=0$, and $pi=1$. Equivalently $p_1[f]=\mathrm{\Delta }(f)e$, where $\mathrm{\Delta }(f)=(f(t_L)-f(t_R))\mathrm{/}(t_L-t_R)$ and $\mathrm{\Delta }(fg)=f(t_L)\mathrm{\Delta }(g)+\mathrm{\Delta }(f)g(t_R)$. For tensor/path generators split the final word as $uzv$ with $u$ nonempty and $z$ one generator, and use $(-1{)}^n[\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}\mathrm{\mid }u\mathrm{\mid }z]v$. The same cancellation works with object idempotents. For a semi-free dg presentation $d{z}_i=f_i$, retain the internal differential by its generator filtration; the corrected small differential is exactly $d(d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}{z}_i)=d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}{f}_i$. The inclusion and homotopy are corrected by that filtration, rather than incorrectly taking every generator to $[z_i]$ when $f_i$ is a word. Each input word involves finitely many corrections. Use a finite perfect representative of the diagonal before dualizing; do not dualize an infinite direct sum termwise. This constructs the complete diagonal and its cyclic dual, including all higher relation rows.

The constructed source has the required CY4 trace and framing, which are verified rather than appended as target assumptions. For the directed category ${\mathcal{D}}_{+}$ write $\mathrm{\Theta }$ for the dual of the diagonal resolution of step 10.1 and $\theta =\mathrm{\Theta }[3]$. Its completion $\mathcal{B}=T_{{\mathcal{D}}_{+}}(\theta )$ has relative diagonal resolution the cone of $\mathcal{B}{\otimes }_{{\mathcal{D}}_{+}}\theta {\otimes }_{{\mathcal{D}}_{+}}\mathcal{B}\to \mathcal{B}{\otimes }_{{\mathcal{D}}_{+}}\mathcal{B}$, whose map is $x\mapsto x\otimes 1-1\otimes x$. Exactness is the same first/last-word cut contraction. The two terms are shifted duals by evaluation of the finite perfect diagonal and its dual, and the displayed difference is the negative shifted transpose of itself. Substitution of that diagonal resolution consequently identifies this completed diagonal resolution with its bimodule dual shifted by four. Tensor-Hom evaluation on proper-support finite families gives the perfect CY4 Ext pairing; exchanging the two factors yields the Koszul sign $(-1{)}^{i(4-i)}$, so it is the cyclic trace used in all the block maps. Setting every old vertex/module to zero and keeping one source line leaves only its scalar degree-zero gauge and its degree-four trace dual in the Hom resolution; hence the original constructed source is spherical. For that source and an old $G$, the forward endpoint row is $R{\mathrm{Hom}}_Y(L_Y,G)$ shifted to Ext degree one, with no source degree-two relation row. The prescribed concentration makes it exactly ${\mathrm{F}\mathrm{r}}_k(G)$ in degree one. Its reverse endpoint is the degree-three trace dual. This establishes the actual original framing identification in the constructed model and does not identify that model with a different source category having the same heart.

Evaluation at the actual spectral action gives a concrete contraction. Put $T=t-\phi$ on $R_F=[A\otimes (P\otimes K_Y^{-1})\to A\otimes P]$. For $f=\sum f_k{t}^k$ define $r(f)=\sum f_k{\phi }^k,q(f)={\sum }_{k\ge 1}{\sum }_{j=0}^{k-1}{f}_k{t}^{k-1-j}{\phi }^j,f=q(f)T+r(f)\mathrm{.}$ Coefficients are never commuted past $\phi$. In the horizontal endomorphism complex of $R_F$, order the source term first: upper $v$ has degree $-1$, diagonal $(a,b)$ degree zero and lower $u$ degree one, with $dv=(vT,Tv)$ and $d(a,b)=Ta-bT$. The small inclusion $\mathrm{\Gamma }$ is the graded diagonal $b\mapsto \mathrm{diag}(b,b)$ plus the lower row $h$. Its differential is the commutator with $\phi$. Set $\pi (v)=0,\pi (a,b)=r(a),\pi (u)=r(u),H(v)=0,H(a,b)=\mathrm{upper}(q(a)),H(u)=\mathrm{diag}(0,-q(u))\mathrm{.}$ The identities $r(Ta)=r(a)\phi -\phi r(a)$ and $q(Ta)=Tq(a)+r(a)$ give $dH+Hd=1-\mathrm{\Gamma }\pi$, $H^2=H\mathrm{\Gamma }=\pi H=0$, $\pi \mathrm{\Gamma }=1$. For a complex $P$, use the tensor totalization signs, multiplying the horizontal $H$ by $(-1{)}^{\text{internal degree}}$. Closedness of $\phi$ makes the internal differential commute with these formulas. These are evaluated UNIVERSAL divided differences from11.2. The reverse endpoint contraction is their derived cyclic dual, fixed before evaluation. The two diagonal traces of $\mathrm{\Gamma }$ cancel and its off-diagonal trace is zero; thus $\tau =\mathrm{tr}H$ satisfies $d\tau +\tau d=\mathrm{tr}$. This is the actual spectral determinant trace homotopy, not a replacement of source endomorphisms by constants.

The construction is independent of its resolving presentation by an actual comparison. For two cofibrant diagonal resolutions take the functorial bar common refinement and their augmentation maps. Since the diagonal is perfect, dualizing their maps computes the same derived dual and gives quasi-isomorphisms between cofibrant representatives of $\mathrm{\Theta }$. A tensor power of these maps is a quasi-isomorphism by induction: each cofibrant factor is homotopically flat and hence preserves the preceding quasi-isomorphism under tensor product. Summing the word lengths gives the quasi-isomorphism of the tensor categories, because direct sums of complexes of modules are exact. On their module categories, extension and restriction of scalars are inverse derived functors: their unit and counit are quasi-isomorphisms on representable modules by the just proved Hom comparison, then on the bar resolutions of arbitrary modules by its tensor filtration and exact direct sums. The comparisons preserve the named original objects and forward evaluations; their opposite, moment and trace maps are the actual common-bar comparisons, not identity maps from an old completed dg subcategory. The diagonal evaluation pairing and its antisymmetric difference map are natural under those refinements, so they preserve the cyclic trace, the actual fibre/cofibre obstruction arrows and ordered determinant trivializations. This supplies resolution-independent constructed source and master models and their coherent overlap maps; no equivalence with an independently chosen bare source is inserted.

We separate the two source reductions. Geometrically let $B=\mathrm{End}(R_F)$ and $K=R\mathrm{\Gamma }({\mathcal{O}}_X)$. The trace fibre of $f(a,b)=a-\mathrm{tr}b$ has coordinates $(a,b,\lambda )$, $\lambda \in K[-1]$, and differential $(da,db,a-\mathrm{tr}b-d\lambda )$. Its contraction to the old small spectral model is $i_{\mathrm{g}\mathrm{e}\mathrm{o}}(c)=(0,\mathrm{\Gamma }c,0),p_{\mathrm{g}\mathrm{e}\mathrm{o}}(a,b,\lambda )=\pi b,H_{\mathrm{g}\mathrm{e}\mathrm{o}}(a,b,\lambda )=(\lambda +\tau b,Hb,0)\mathrm{.}$ Direct substitution using12.2 gives $d{H}_{\mathrm{g}\mathrm{e}\mathrm{o}}+H_{\mathrm{g}\mathrm{e}\mathrm{o}}d=1-i_{\mathrm{g}\mathrm{e}\mathrm{o}}{p}_{\mathrm{g}\mathrm{e}\mathrm{o}}$ and the three side conditions. In particular the function-valued source background is removed through the ACTUAL trace fibre; it is not identified with $\mathbb{C}$ or $\mathbb{C}[-4]$. On the completed side its common central identity and cyclic trace-dual are removed separately. The central identity is $({\mathrm{id}}_{\mathrm{o}\mathrm{l}\mathrm{d}},{\mathrm{id}}_p)$, not a pure source scalar, which would fail to commute with $s$. In its gauge coordinates put $\beta =b-a{\mathrm{id}}_{\mathrm{o}\mathrm{l}\mathrm{d}}$; eliminate that central direction and its cyclic dual by their two identity mapping-cone pairs. The unperturbed reduced old row is then the same small spectral model $\mathrm{\Gamma }$, its forward endpoint is the actual framing complex, and its reverse is the cyclic dual fixed in12.2. The coevaluation trace and the homotopy $\tau$ give these reductions their common endpoint comparison; no equality between the two unreduced source backgrounds has been used.

The completed and geometric endpoint maps can now be checked on matrix units, before transfer. A directed row has boundary $z\otimes 1-1\otimes z$. Its evaluated left and right actions are composition, and contraction of its dual with an endomorphism is ${\sum }_{\alpha }(-1{)}^{\mathrm{\mid }{e}_{\alpha }\mathrm{\mid }}{b}_{\alpha \alpha }={\mathrm{tr}}_{P}b$. Therefore, for $I=\mathcal{O}\oplus Q$, $Q=P_F[-1]$, $D=\mathrm{diag}(d_{\mathcal{O}},d_Q)+S$ and $S_{21}=s$, a degree-$n$ matrix has $[D,(a,v;u,b)]=(da-(-1{)}^{n}vs,dv;du+sa-(-1{)}^{n}bs,db+sv)\mathrm{.}$ Both endpoint moment products $vs,sv$ occur; cyclicity gives ${\mathrm{tr}}_P(sv)=-(-1{)}^{n}vs$. The trace-kernel substitution is exactly $a={\mathrm{tr}}_{P}b$, giving $j,q$ of6.1. Thus the dual endpoint has not been set to zero or inferred from its rank. Under the common uncontracted diagonal of11.2, adding the actual section is on both sides the SAME endpoint operator $\delta =[S,-]$: this is the evaluation of its two boundary terms, and its shifted transpose is fixed by the dual diagonal. It sends reverse to diagonal to forward to zero. Let $(i,p,H)$ be the unperturbed contractions12.2 and 13.2 together with the forward one and its cyclic dual. These preserve endpoint blocks, so $(H\delta {)}^3=(\delta H{)}^3=0$. Transfer is the finite calculation $i_s=i-H\delta i+H\delta H\delta i,p_s=p-p\delta H+p\delta H\delta H,d_s=d_0+p\delta i-p\delta H\delta i+p\delta H\delta H\delta i,$ with $H_s=H-H\delta H+H\delta H\delta H$. Substitution proves the chain and contraction identities. On the completed diagonal these are the word-cut contraction and its actual dual evaluated at $s$; on the geometric cone they are the same contraction of $[D,-]$. Consequently their transferred differential is this SAME $d_s$, including the corner path through the old row and the trace-homotopy contribution from13.2. Transfer the trace fibre and $\tau$ with these formulas and then apply $j,q$. Writing $B,G$ for the completed and geometric contractions to this common transferred complex, the ACTUAL endomorphism-complex maps are $f=i_{G,s}{p}_{B,s}$ and $g=i_{B,s}{p}_{G,s}$. Their composites are homotopic to the identities by the displayed $H_s$ and its transferred comparison. The required obstruction-complex maps are $g^{\vee }[-1]$ and $f^{\vee }[-1]$. Thus this gives actual inverse cochain comparisons with $R{\mathrm{Hom}}_X(I,I{)}_0^{\vee }[-1]$, not an unspecified isomorphism of its cohomology. All syzygies are still present in the uncontracted diagram; the comparison is not an assertion from matching graded blocks.

The global comparison has explicit descent homotopies. Use the SAME sheaf normalized bar on all charts, containing the actual universal family, its diagonal evaluation, trace and universal derivation. For local reductions $(i_a,p_a,h_a)$ to it define $t_{ba}=p_b{i}_a$. On a triple overlap $t_{cb}{t}_{ba}-t_{ca}=-d(p_c{h}_b{i}_a)-(p_c{h}_b{i}_a)d\mathrm{.}$ Higher overlap maps are the signed strings $p_{a_n}{h}_{a_{n-1}}\cdots h_{a_1}{i}_{a_0}$. Their Hom differential is ${\sum }_{j=1}^{n-1}(-1{)}^{n-1-j}{p}_n{h}_{n-1}\cdots h_{j+1}(1-i_j{p}_j)h_{j-1}\cdots h_1{i}_0\mathrm{.}$ The $1$ terms delete an intermediate index and the $ip$ terms split the string, exactly proving every higher cocycle equation. Totalize these actual Čech/atlas diagrams, retaining their positive Čech components. The same strings give inverse total comparisons with the common bar. The trace and pairing use its cyclic dual and $\tau$, so the identical prism calculation preserves them, not merely their fibrewise cohomology forms. For an unbounded bar use its filtered derived realization before a finite perfect replacement; no termwise dual of an infinite sum or commutation of arbitrary derived tensor with an infinite totalization is needed. The resulting map is a map of perfect sheaf complexes on the original $Z_X$. Applying derived pullback to THAT map preserves its identities; locally it is exactly the polynomial, endpoint and prism formulas above. This supplies coherent refinement, named-arrow and direct-sum comparisons without a globally split old/relative cotangent decomposition.

We now compare the ORIGINAL universal module/pair Atiyah cocycles. Let ${\mathcal{C}}_B$ be the source-reduced complex $R{\mathrm{Hom}}_{\mathcal{B}}(M,M)$ of the original universal completed module and let ${\mathcal{C}}_G=R{\mathrm{Hom}}_X(I,I{)}_0$. The maps $f=i_G{p}_B$ and $g=i_B{p}_G$ are those of14.1, with the separate source reductions13.2. On an affine/Čech chart of the ORIGINAL $Z_X$, use a cofibrant coefficient algebra $R$ resolving its structure sheaf and ${\mathrm{\Omega }}_R$ with its differential as the model for its FULL cotangent complex. For the actual MODULE bar resolution $Q_M={\mathrm{Bar}}_{\mathcal{B}}(M)$, its differential $D_{Q_M}$, augmentation ${ϵ}_M$ and coefficient connection $\mathrm{\nabla }$, the principal-parts convention of the original arrows is represented by ${\kappa }_B={ϵ}_M[D_{Q_M},\mathrm{\nabla }]\in R{\mathrm{Hom}}_{\mathcal{B}}(M,M){\otimes }^L{L}_{Z_X}[1]\mathrm{.}$ The connection differentiates the coefficient algebra, fixing the category and $Y$ directions, and the target differential includes $d_{{\mathrm{\Omega }}_R}$. This is the cocycle of the ORIGINAL MODULE, before forming its endomorphism comparison complex. In a strict dg-module bar model its length-zero entry is $-\mathrm{\partial }{D}_M$ and its length-one entry is the derivative of the actual terminal action term, $-\mathrm{\partial }{\rho }_M(z)$ with the bar suspension sign. Higher normalized bar lengths are zero in this strict model. An algebra word inside one entry is differentiated at EVERY letter by the graded Leibniz rule; every negative relation generator is retained. If a coherent or $A_{\mathrm{\infty }}$ presentation is used first, its higher actions contribute their derivatives, and an actual MODULE bar strictification with the module comparison below is used. The same formula on a total twisted resolution includes derivatives of every transition and higher Čech entry. For the geometric side use the ORIGINAL module resolution $R_F=[A\otimes (P\otimes K_Y^{-1})\stackrel{t-\phi }{\to }A\otimes P]$ and $P_I=(L_p\otimes {\mathcal{O}}_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}})\oplus R_F[-1]$, with its actual total module differential $D_I$. Both induced copies of $P$ use the SAME parameter connection and the SAME transition/higher-transition data; $t,K_Y$ and the fixed $X$-line have zero parameter derivative. Put ${\kappa }_I=[D_I,{\mathrm{\nabla }}_I]\in R{\mathrm{Hom}}_X(I,I){\otimes }^L{L}_{Z_X}[1]\mathrm{.}$ Its old diagonal is the two shifted copies of ${\kappa }_P=[D_P,{\mathrm{\nabla }}_P]$; its spectral off-diagonal entry is $+\mathrm{\partial }\phi$ from $-\mathrm{\partial }(t-\phi )$; its primary forward entry is $-\mathrm{\nabla }s$; the reverse entry is zero. Here $\mathrm{\nabla }s={\mathrm{\nabla }}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}s-s{\mathrm{\nabla }}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}$. These are derivatives of the ORIGINAL cone MODULE differential, not derivatives of $[D_I,-]$ on its endomorphism complex. On an overlap the same formula marks the actual transition or higher twisting map, with its total sign. Those entries need not vanish. Retain the source line's full cocycle $a_p=\mathrm{At}(L_p)$, which may have nonzero positive Čech components even when local frames have zero connection. The ordered induced resolution gives $\det F={\pi }^{\ast }{K}_Y^r$ for $r=\mathrm{vrk}P$; it is a FIXED $X$-line along the parameter, not necessarily the trivial line. In the compatible induced presentation its relative trace cocycle cancels term by term between the two copies of $P$, while all Higgs and section derivatives are off-diagonal. Hence $\mathrm{tr}{\kappa }_I=a_p$. The geometric source-normalized class is the ACTUAL trace-fibre cocycle ${\kappa }_I^0=({\kappa }_I-a_p{\mathrm{id}}_I,{\lambda }_{\det }),d(x,c)=(d_{\mathrm{E}\mathrm{n}\mathrm{d}}x,\mathrm{tr}x-dc)\mathrm{.}$ The determinant identification specifies ${\lambda }_{\det }=0$ in these compatible total models. A connection change carries its scalar primitive as well; it is not erased. On the completed side subtract the SAME $a_p$ times the common central identity before the gauge/cyclic reduction13.2. This acts on both old and source entries, not on a pure source scalar. Denote that original reduced module cocycle by ${\kappa }_B^0$. No scalar complex is identified with $R\mathrm{\Gamma }({\mathcal{O}}_X)$. The images of these ORIGINAL marked cocycles can be calculated explicitly. The module-bar cochain is evaluated on the inclusion of a small generator $z$: its leading term is $[z]$ and its internal-differential corrections have bar length at least two. The latter give zero against the strict length-zero/one module marking. The Leibniz identity of11.2 therefore sends it to the actual coefficient derivative of the action at that generator, including all higher syzygy actions. The right-module bar boundary of the polynomial generator is action-minus-free, $\phi -t$; the identification with the induced resolution $t-\phi$ reverses this generator's sign. Thus its marked entry is exactly $+\mathrm{\partial }\phi$, while the primary forward entry is $-\mathrm{\nabla }s$. The module differential contributes the two copies of ${\kappa }_P$. Consequently BOTH projections give the same source-normalized marking

Every marked entry is constant in the spectral polynomial variable $t$, although it can have any coefficient-dg or Čech degree. Hence $q(\text{constant})=0$ in12.2, the old and forward homotopies annihilate the marking, and the reverse marked entry is zero. The section perturbation keeps these marks in the same polynomial-constant diagonal/forward positions. Therefore $H{\kappa }_0=H\delta {\kappa }_0=0$: the finite HPL corrections vanish ON THIS ORIGINAL MARKED COCYCLE, even though they are nonzero on general cochains. This proves the displayed image equality and $i_G{\kappa }_{\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}}={\kappa }_I^0$ in compatible induced models. It does not differentiate an endomorphism-complex contraction. The calculation includes each positive Čech component: a mark on a transition or higher homotopy is copied in the same two induced terms, and a marked section/Higgs homotopy remains its actual forward/off-diagonal coefficient. Common bar refinements compare those SAME marked words. Here is the explicit homotopy for arbitrary allowed resolution/connection choices. For a comparison $U:P\to Q$ of resolutions of the SAME original module, let $V$ be its inverse and $d_{\mathrm{H}\mathrm{o}\mathrm{m}}h=1-UV$. Its MODULE connection defect is $K_U={\mathrm{\nabla }}_{Q}U-(U\otimes 1){\mathrm{\nabla }}_P$, so direct expansion gives ${\kappa }_{Q}U-(U\otimes 1){\kappa }_P=d_{\mathrm{H}\mathrm{o}\mathrm{m}}{K}_U,{\kappa }_Q-U{\kappa }_{P}V=d_{\mathrm{H}\mathrm{o}\mathrm{m}}(K_{U}V-{\kappa }_{Q}h)\mathrm{.}$ This formula is applied to the original resolutions of $M$ or $I$, never to resolutions of ${\mathcal{C}}_B$ or ${\mathcal{C}}_G$. For a connection change $B={\mathrm{\nabla }}_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{n}}-{\mathrm{\nabla }}_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}\mathrm{e}\mathrm{d}}$, the marking changes by $d_{\mathrm{H}\mathrm{o}\mathrm{m}}B$; use $B-B_p\mathrm{id}$ after source normalization and retain its trace-fibre lift $d(B,0)$. These explicit module primitives and the common module-bar refinement prisms define ${\alpha }_B,{\alpha }_G$ with ${\kappa }_B^0={\kappa }_{B,\mathrm{a}\mathrm{d}}^0+d{\alpha }_B$ and ${\kappa }_I^0={\kappa }_{I,\mathrm{a}\mathrm{d}}^0+d{\alpha }_G$. All expressions are in the total coefficient/Čech Hom complexes, with their scalar trace components. Put $t=p_B{\alpha }_B-p_G{\alpha }_G,{\eta }_G=i_{G}t-H_G{\kappa }_I^0\mathrm{.}$ The compatible marked equality above and $d{H}_G+H_{G}d=1-i_G{p}_G$ give the REQUIRED identity of ORIGINAL classes $p_B{\kappa }_B^0-p_G{\kappa }_I^0=dt,(f\otimes 1){\kappa }_B^0-{\kappa }_I^0=d{\eta }_G\mathrm{.}$ The inverse comparison has ${\eta }_B=-H_B{\kappa }_B^0-g{\eta }_G$, so $(g\otimes 1){\kappa }_I^0-{\kappa }_B^0=d{\eta }_B$. These are homotopies in the original module/pair deformation complexes tensored with $L_{Z_X}$, including the trace-fibre scalar entry. They are not homotopies in $\mathrm{End}({\mathcal{C}}_{\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}})$. The identities descend using the SAME module bars and the prism strings15.1. Apply the MODULE connection-defect identity to their actual module comparison maps and higher homotopies, or compute it once in the full Čech total complex; this supplies the displayed $\alpha$ and $\eta$ in every Čech degree. For independent secondary sources keep the original connection/Atiyah markings of EVERY auxiliary vector bundle, flag and sink in the Artin gauge rows, and every derivative of its named arrow and resolved relation. Only the primary common-central $a_p$ is removed, tensorially at every named vertex: an initially unnormalized auxiliary $V_a$ retains $\mathrm{At}(V_a\otimes L_p^{\vee })=\mathrm{At}(V_a)-a_p{\mathrm{id}}_{V_a}$, rather than a zero mark. These remaining relative marks are polynomial-constant, so the OLD spectral division homotopy annihilates them. For the $bc$/dual elimination we identify the ORIGINAL modules and maps explicitly. Work over the SAME cofibrant coefficient algebra $R$ of the original family and SAME full completed category, written ${\mathcal{A}}_R$ in left-module convention (the opposite of the right-module convention of10.1). Its original universal module $M$ is the one induced by the SAME dg quotient11.1. The old spectral action is unchanged and can be nonzero; its new dual/moment values are zero, but their full bar deformation rows remain. At the three named vertices write $E=M_{l-1}$, $F=M_l$, $W=M_0$. The actual actions satisfy $d_{\mathrm{H}\mathrm{o}\mathrm{m}}r=bc$. On an injective-$c$ chart invert a maximal minor and choose a CLOSED $\mathrm{\ell }:F\to E$, $\mathrm{\ell }c=1$. Define an actual ${\mathcal{A}}_R$-module $N$ on the SAME vertex complexes by $c_N=c$, $b_N=b-(d_{\mathrm{H}\mathrm{o}\mathrm{m}}r)\mathrm{\ell }=b(1-c\mathrm{\ell })$, $r_N=0$, all other actions unchanged. Then $b_{N}c=0$, so this is a dg module. In adapted coefficient coordinates $S=S_0[y,r]$, $dr=y=bc$, it is the reduced $S_0$ diagram pulled to the given $R$; no coefficient-ring contraction is used to infer a module equivalence. Define the degree-$-1$ twisted derivation $H$ by $H(b)=-r\mathrm{\ell }$, zero on EVERY other generator, and $H(uv)=H(u){\rho }_M(v)+(-1{)}^{\mathrm{\mid }u\mathrm{\mid }}{\rho }_N(u)H(v)$. On $b$, $dH(b)=-bc\mathrm{\ell }={\rho }_N(b)-{\rho }_M(b)$; on $r$, $H(dr)=H(bc)=-r\mathrm{\ell }c=-r={\rho }_N(r)-{\rho }_M(r)$. On every remaining generator the identity ${\rho }_N-{\rho }_M=dH+H{d}_{\mathcal{A}}$ is zero on both sides, including new dual/moment generators by the SAME dg quotient11.1. A word containing a new dual/moment has $H=0$ by this twisted Leibniz rule and its zero evaluated action. Every nonzero $H$-value is a directed path ending in $W$, zero on input $W$. Hence $H(u)H(v)=0$ even on the FULL completed bar: formal reverse paths have not been forbidden, but their intervening evaluated dual actions are zero. Put $P={\mathrm{Bar}}_{{\mathcal{A}}_R}(M)$ and $Q={\mathrm{Bar}}_{{\mathcal{A}}_R}(N)$, the actual normalized MODULE resolutions. With $\mathrm{\mid }[a]\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }-1$ and terminal bar differential $D(1[a]m)=a\otimes m-1\otimes \rho (a)m-1[da]m+(-1{)}^{\mathrm{\mid }a\mathrm{\mid }-1}1[a]d_{M}m$, let $T_H$ consume the last bar entry: $T_H(a_0[a_1\mathrm{\mid }\cdots \mathrm{\mid }{a}_n]m)=a_0[a_1\mathrm{\mid }\cdots \mathrm{\mid }{a}_{n-1}]H(a_n)m(n\ge 1),T_H(a_{0}m)=0.$ The map $[a]m\mapsto H(a)m$ has degree zero, fixing the suspended tensor signs. Set $U=1+T_H:P\to Q$, $V=1-T_H:Q\to P$. The length-one chain-map equation is precisely ${\rho }_N-{\rho }_M=dH+H{d}_{\mathcal{A}}$; the length-two equation is the displayed twisted derivation identity; every earlier face cancels unchanged. Thus $D_{Q}U=U{D}_P$ in every bar length and internal degree. Since $T_H^2=0$, one has $VU=1_P$, $UV=1_Q$ EXACTLY and $h_P=h_Q=0$. These are original-module maps over the same ${\mathcal{A}}_R,R$, not endomorphism-complex maps promoted to module maps. Use the SAME underlying vertex connections on $P,Q$. For the convention $\kappa =[D,\mathrm{\nabla }]$ above, their concrete defect is $K={\mathrm{\nabla }}_{Q}U-U{\mathrm{\nabla }}_P={\mathrm{\nabla }}_{\mathrm{H}\mathrm{o}\mathrm{m}}{T}_H$, with terminal $b$ entry ${\mathrm{\nabla }}_{\mathrm{H}\mathrm{o}\mathrm{m}}(-r\mathrm{\ell })$ and its word derivatives. It retains the derivative of $\mathrm{\ell }$ and every vertex connection. The same endpoint support gives $K{T}_H=0$, so $KV=K$ and ${\kappa }_Q-U{\kappa }_{P}V=d_{\mathrm{H}\mathrm{o}\mathrm{m}}K$; the previously generic $K_{U}V-{\kappa }_{Q}h$ is THIS $K$, with $h=0$. Its scalar trace-fibre component is explicitly ZERO. Indeed $U$ is identity on every vertex and on the original primary pair $I$; $K$ has only a strictly forward-to-$W$ endpoint corner and no primary-source/old diagonal component. In the finite endpoint/cyclic model12.2 and 13.2 its evaluation against an identity coevaluation is zero by the distinct endpoint idempotents; any closing new-dual action is zero. Thus the lift is $(K,0)$, with $d(K,0)=(d_{\mathrm{E}\mathrm{n}\mathrm{d}}K,0)$. The actual $a_p$, original determinant identification and ${\lambda }_{\det }$ are unchanged. This computes the scalar component in the finite trace model, not by tracing the infinite bar. Auxiliary residual Atiyah marks remain. Conjugation $T\mapsto UTV$ transports the FULL endomorphism complex; its cyclic-dual rows use the inverse transpose maps and retain evaluation since $VU=UV=1$. Here is its positive-Čech application without an assumed horizontal splitting. For the ACTUAL original module twisting entries $G_{\sigma }:P_{a_k}\to P_{a_0}$ of15.1, of degree $1-k$, put $G_{\sigma }^{\mathrm{\prime }}=U_{a_0}{G}_{\sigma }{V}_{a_k}$ for EVERY $\sigma =(a_0,\dots ,a_k)$. All higher module cocycle equations follow by cancelling the exact $V_a{U}_a=1$ in every product. On a same-frame splitting overlap $U_{{\mathrm{\ell }}^{\mathrm{\prime }}}{V}_{\mathrm{\ell }}=1+T_{H_{{\mathrm{\ell }}^{\mathrm{\prime }}}-H_{\mathrm{\ell }}}$, with $(H_{{\mathrm{\ell }}^{\mathrm{\prime }}}-H_{\mathrm{\ell }})(b)=-r({\mathrm{\ell }}^{\mathrm{\prime }}-\mathrm{\ell })$; its triples telescope strictly. With the actual bundle/resolution transitions use the stated $G_{\sigma }^{\mathrm{\prime }}$, retaining every higher entry. In total notation $D_Q^{\mathrm{t}\mathrm{o}\mathrm{t}}=U{D}_P^{\mathrm{t}\mathrm{o}\mathrm{t}}V$. Covariantly differentiating the actual original-module entries gives, in EVERY Čech degree, $(\mathrm{\nabla }{G}_{\sigma }^{\mathrm{\prime }})U_{a_k}-U_{a_0}(\mathrm{\nabla }{G}_{\sigma })=K_{a_0}{G}_{\sigma }-G_{\sigma }^{\mathrm{\prime }}{K}_{a_k},\mathrm{\nabla }{V}_a=-V_a(\mathrm{\nabla }{U}_a)V_a\mathrm{.}$ These are exactly the positive-Čech components of ${\kappa }_{Q}U-U{\kappa }_P=d_{\mathrm{H}\mathrm{o}\mathrm{m}}K$; transition derivatives are not zeroed. Connection changes of the original bundles remain on BOTH sides, and the new covariant defect still has the displayed forward support, hence scalar entry zero in every Čech degree. This constructs the descended original-module primitive $(K,0)$. Its images under the already specified bar/spectral/source projections are the concrete secondary contributions to ${\alpha }_B,{\alpha }_G$, and thence to $t,{\eta }_G$ above. Use the finite acyclic forward filtration and degreewise word cuts of11.2 and14.1, not the primary two-insertion bound for an entire flag. Thus the same marked-diagram calculation supplies the original secondary universal-arrow square of17.1; it does not impose a new PVP premise. Finally the original obstruction arrows are the adjoints of THESE cocycles: ${\varphi }_B:{\mathcal{C}}_B^{\vee }[-1]\to L_{Z_X}$ and ${\varphi }_G:{\mathcal{C}}_G^{\vee }[-1]\to L_{Z_X}$. The identity for ${\eta }_G$ gives ${\varphi }_B{f}^{\vee }[-1]\simeq {\varphi }_G$, and that for ${\eta }_B$ gives ${\varphi }_G{g}^{\vee }[-1]\simeq {\varphi }_B$. This is the original universal cotangent square, not the Atiyah class of an endomorphism complex. For every derived $R^{\mathrm{\prime }}\to Z_X$, pull back the entire cocycles, determinant primitive, maps and these homotopies FIRST to the pullback of the FULL $L_{Z_X}$; polynomial division, module bar operations and the coefficient/Čech identities commute with that derived tensor. Proper-support perfect duality gives the same trace. Only then compose with its canonical map to $L_{R^{\mathrm{\prime }}}$. No smaller base, restricted module class or absolute-POT assertion after arbitrary base change is used.